Monday, July 23, 2007

HELIOS

Hydrogen nucleons fuse,
Explode over billions of years
Like a fusion, the sum of all farts;
In part the sunny signature
Of its maker, like a hearth’s heat
Suffusing the home of our hearts.

Saturday, July 21, 2007

Of Mice and Mengen

As mentioned in my last post, the Burali-Forti paradox seems to be, from the standard perspective (see the aforementioned paper), the most fundamental and paradoxical of the set-theoretical paradoxes, although from my own perspective it is the least fundamental, which is why I’ve already looked at Russell’s and Cantor’s paradoxes.
......I’ll just briefly describe the standard paradox, which arises from the standard presupposition that the natural numbers are a standard set, that they form an actual (or finitesque) totality—in more picturesque terms, one that can, when they (or rather, their tokens) are envisaged spatially, be thought of as being all together, in the same space; or one that can, when their endless sequence is thought of more temporally (in the intuitive sense, i.e. not as part of a quasi-spatial space-time), be completely run through. (For an independent, scientific reason why that standard presumption is likely to be false, see my forthcoming paper.)
......Beginning with standard set theory and its von Neumann ordinals (0 = the empty set, 1 = {0}, 2 = {0, 1}, 3 = {0, 1, 2}, and so forth), we have that each ordinal is the set of the preceding ordinals. The first transfinite ordinal is omega = the set of all the natural numbers, and it is followed by omega + 1 = {0, 1, 2, …, omega}, and so on. There is no set of all such ordinals because if there were it would similarly be an ordinal, and it would be greater than (since arising higher in the set-theoretic hierarchy than) all its members, whereas they were presumed to be all such ordinals. Paradox arises because our intuition that there should be a set of all the natural numbers (and hence our first transfinite ordinal) seems also to tell us that there should be such a set of all those ordinals (and hence some last ordinal, usually denoted by capital-omega).
......Cantor originally needed transfinite inductions, and hence his set (Menge) theory, because he was investigating arbitrary functions of the real number line, as part of the development of Fourier analysis. Although the functions usually met with in physics are well behaved (smooth, differentiable etc.) almost everywhere, one generally expects there to be points (i.e. real numbers) where they are discontinuous etc. Analysis is simplest when there are not infinitely many such points in any finite interval, but if there are then they will have at least one accumulation point, and there might be infinitely many such accumulation points, with accumulation points of their own, etc.
......Cantor was therefore considering sequences of point-sets, each set of points obtained from the preceding one by keeping only the accumulation points. Often, for a given function, the Nth point-set in that sequence would, for some natural number N, contain only a finite number of points, with the (N + 1)th point-set being empty, but the particularly troubling (and therefore mathematically interesting) cases were the functions for which the Nth point-set contained infinitely many points for every natural number N (e.g. the function whose value is 1 at each rational point, and 0 elsewhere). So Cantor was considering what was left after all the naturally numbered stages of that process had been gone through, i.e. at the omegath stage, and (since that might also contain infinitely many points) beyond.
......And one reason why the Burali-Forti paradox remains so paradoxical (see the aforementioned paper) is that today’s mathematicians seem to find a use for inductions that range over ordinals that are (in some mysterious way) even bigger than capital-omega, e.g. in mouse theory. So, what got us using standard ordinals in the first place does seem to take us beyond them, paradoxically. And so perhaps the epistemic possibility that the natural numbers do not actually form a set (which does not presuppose a constructivistic view of numbers, but only that they are an objective structure, whose properties are to be discovered rather than stipulated) should not be too casually dismissed.
......If the natural numbers happen not to form a set, but are rather only potentially infinite, then although we could still consider ordinals insofar as they are the order-types of well-orderings (i.e. as reifications of certain structures) we would not have the same use for them. Cantor’s naturally numbered stages could not have been completely run through, to get us to an omegath stage, but that would not matter because geometrical lines would not then be isomorphic to real number lines, so there would not be such arbitrary functions to investigate in the first place.
......That is not to say that investigating them was not useful, that the standard approach has not given us a nice formal model of geometry, adequate for most applications, but only that it would then be (as it seems to be) no more than that. Note that Cantor did dismiss that possibility rather casually—remarking that a potential infinity (thought of as a variable) presupposes a logically prior actual infinity (the domain of the values that the variable could take)—perhaps too casually (or Hegelian?) given that he was aware of these paradoxes.

Monday, July 16, 2007

More Carnivals

The day of the 50th Philosophers' Carnival, and time to use the carnival submission form to tell me of interesting posts related to the philosophy of mathematics (or the philosophies of logic and science) and mathematical philosophy, which is the use of mathematics (including formal logics, probabilities and scientific models) to tackle any philosophical topic.
......I don't want to exclude any interesting stuff, so any recent post will be considered, but the more mathematical it is the more extensively will “recent” be taken, and the less exceptionally interesting it will have to be. Your post doesn't need to be anything earth-shattering - it just needs to be something that other philosophically-minded people might enjoy reading.
......At the moment I'm thinking about the Burali-Forti paradox, as I'm reading All Things Indefinitely Extensible (from the book being reviewed over on Logic Matters). (PS, July 19 sees the 12th Philosophia Naturalis.)

Friday, July 13, 2007

Metaphysical Progress

Heraclitus: Everything is made of water, and the world is full of gods. Thousands of years later and we reach David Lewis: Everything supervenes upon the local properties of space-time points, and the worlds are full of miracles. Hmm…

Tuesday, July 10, 2007

Physics and Ethics

This post is a bit long (fourteen hundred words) because I’m asking a question that strikes even me as odd—and if I seem to be defending the popular answer, that is only to emphasise that there is a genuine question here—i.e. who are the experts on ethics? For an example of why that may not seem to be much of a question, consider Anthony Grayling’s commonplaces about Faith (as opposed to science) in his The Meaning of Things (page 122):
......“Contemporary science hypothesises an evolutionary tale of
......physical forces. I say ‘hypothesises’, note; hypothesises on
......the basis of good evidence, severely tested, with many
......aspects of the accompanying theory successfully applied to
......daily life – as exemplified by the light you read by, the
......computer you work on, the airplane you fly in. The great
......advantage of science’s careful and thorough hypotheses,
......always ready to yield if better evidence comes along, is that it
......makes use of no materials or speculations beyond what the
......world itself offers. Religions, in sharp contrast, offer us eternal
......certitudes on the basis only of ancient superstition.”
But to begin with, surely our sciences do not only utilise “what the world itself offers” us—e.g. for most people (including most scientists) the standard model of those physical forces is taken on trust from particle physicists—not, that is, unless such things as the reputations of such people are included within “the world,” as perhaps they ought to be since we are, each of us, part of, rather than apart from, the world—although incidentally, if they are included then religious beliefs, about the non-physical aspects of the world, might not be so very different in kind from our scientific beliefs, about its physical aspects (cf. Polkinghorne’s defence of the Trinity as the best explanation of our written records; with which, incidentally, I disagree). Anyway, particle physicists base their theories upon data taken (almost exclusively) from places that hardly anyone else could gain admittance to—or could understand the workings of (in order to check what was going on) if they did. I’m not suggesting that such people are not to be trusted, just observing that trusting their authoritative testimony is what we do.
......In fact, although the daily life and work of most scientists (and many others) may well provide plenty of support for our beliefs about chemistry and electromagnetism (and thence in quantum mechanics), this “evolutionary tale of physical forces” is hardly so straightforward. For starters, alternative hypotheses are not always eliminated on the basis of the evidence, but surprisingly often because they are just uninteresting, for various reasons—for a glimpse of the sheer range of unconsidered possibilities, consider the following. Particle (or high-energy) physics research has surely (for the last half-century, at least) been of great interest to the military wings of the great world powers, who of course believe that it is of the utmost importance that certain classes of potential discoveries run little risk of falling into the hands of potential enemies—so my question is, what evidence does the world itself offer that the standard model of particle physics is not a fabrication designed purely for public (and thence potential enemy) consumption?
......Now, that particular hypothesis is not to be taken seriously, of course (it is not in capital letters, on a more popular blog), but clearly there are many possibilities, of various kinds, some quite reasonable (e.g. some non-string-theoretic approaches to particle physics may have been neglected for socio-economic reasons), and my point is not that particle physicists are not (or should not be) the experts about physical forces—far from it—but rather that few sorts of knowledge do not require a considerable degree of faith. There may be some, e.g. elementary mathematics, and our everyday knowledge of the world, and of our own minds, but those are hardly examples of scientific knowledge. Now, I personally favour a hitherto unconsidered (and hence unfalsified) hypothesis about physical continua (see my web pages) over the standard set-theoretic hypothesis—and for such reasons I’m undecided about physics beyond elementary quantum mechanics—but even I recognise that it is important what the experts on mathematics and physics favour (for what are presumably good, if not overwhelming reasons). But then similarly, I’m thinking, surely it is important what the experts on ethics think—whether we like what they say, or not—about what the moral facts are (e.g. that would be important in law and politics). So the question is, who are those experts?
......Similarly, Stephen Hawking is clearly an expert on physical matters irrespective of whatever I think about strings (unfortunately). My worry is that many Naturalists are simply supposing that, just because they believe (on the basis of relatively little evidence at present) that moral facts would, if they exist at all, be best explained by an evolutionary tale, therefore religious thinkers are not experts on ethics. Naturally we think that we know what is right, independently of those religious systems that we are not part of, and of course we do not need those systems in order to function socially (at least not in the short term)—but then, we also know about the physical world independently of the hypotheses of the physicists; and most of us are physically competent, athletes and craftsmen more so than most physicists, who are nonetheless the experts on physics (for good reasons). And since I mean ethics and not meta-ethics (much as I meant above, physics and not metaphysics), so the experts are unlikely to include many philosophers—what philosophy has shown (e.g. via the paradoxes of the trolley and of distance) is that the activity of our consciences no more eliminates our need for experts on ethics than our physical intuitions eliminate our need for physicists.
......But I’m no expert, and maybe I’ve made the question clear enough, so I’ll return to reading Grayling, who is even better than Russell... But on page 100: “The religious attitude is marked by a robust refusal to take things at face value if inconvenient.” That proposition would be even truer were ‘religious’ replaced by ‘naturalistic’ (think of your own feelings and choices, and the physicists’ 10-dimensional strings, etc.)... And on page 101: “Why can we not be prompted to the ethical life by our own charitable feelings? The existence of a god adds nothing to our moral situation,” which inspires me to suggest sarcastically that surely we know about the physical world by bumping around inside it, that surely the existence of strings adds nothing to our physical situation! That is, I could say much the same for mathematics and the existence of sets, or physics and the existence of strings—but even I recognise that if strings exist then they would explain not only how we bump into things—how we actually bump into them, irrespective of what we think we are doing—but also the finer (and more paradoxical) details of the world around us. (And of course, even if they don’t exist, they may—or may not—be a good way of modelling reality.)... Etc.

Saturday, July 07, 2007

An Empty Set?

My problem with singletons was (as previously mentioned) that although it is intuitively clear how a plurality (many things) could also be a unity (one plurality)—e.g. one deck of 52 cards—it is consequently obscure how a single object could similarly be a unity different from that object. Nonetheless there is a related obscurity with pluralities, e.g. how do two pairs of socks differ from those four socks?
......Although our 52 cards (think of any particular deck) are exactly the same as those 4 suits (hearts, clubs, diamonds, spades) of 13 cards each—an application of 4 times 13 being 52—those are two different pluralities (52 cards and 4 suits), two different numbers of (different sorts of) things. In one sense they are the same thing, the same deck, but in another sense they are two different things, two different ways of partitioning the cards. When we refer to a collection—e.g. to a par of socks—we are usually referring not to that partition but to those things—to those two socks—but still, two pairs of socks are one collection, while the same four socks are another.
......Perhaps that is why collections are regarded as abstract objects, because they are akin to partitions (which are clearly abstract objects). And when we compare partitions it makes sense to compare partitions of some given things, so perhaps we should be thinking of partitions of some totality of things that are not collections, some universe (of discourse). After all, it remains fairly mysterious what something being a thing (standing apart from everything else) amounts to, but presumably whatever it is will involve those other things (that other stuff) to some extent.
......Of course, such partitions are just collections of collections, so I’m presupposing collections (of many into one), but if collections are indeed akin to partitions then that might shed some light on singletons—e.g. were each collection all the parts but one of some such universal partition, with everything else (in the universe) in the remaining part, so that a singleton would be a part of a partition. Or perhaps the singleton would be the whole partition (an abstract object), containing its single member (that part of the partition) within the rest of itself (the set-theoretic lasso).
......Now, so far I’ve overlooked overlaps, e.g. the collection of pairs (two aces, two fours etc.) is not a partition of our 52 cards, but of three identical decks—or rather (that being impossible) I should not have excluded collections from the universe. Things would get pretty complicated were they all included to begin with, but something like the standard cumulative-hierarchical approach might work out; and if so then maybe the (non-formal) empty set would be the degenerate partition of (i.e. nothing but the lasso that is) some such cumulative-hierarchical universe?

Thursday, July 05, 2007

No Use?

I probably won't be posting much until I start my PhD in September (elsewhere I've noticed some interesting answers to the question What use are philosophers of mind?) although I'll be hosting the Philosophy Carnival on August 6 (with a logical/mathematical/scientific theme).